Problem Set 7
1. (A variant on the definition of exterior measure). For a cube Q ⊂ Rd of side
length S > 0, define
Cr (Q) := S r .
For a subset E ⊂ Rd , define
Mr (E) :=
inf
E⊂∪∞
j=1 Qj
∞
X
Cr (Q).
j=1
If r = d, them Mr is exactly the exterior measure m∗ .
If r > d, show that Mr ([0, 1]d ) = 0.
If 0 < r < d, show that Mr ([0, 1]d ) = 1.
2. If m∗ (E) = 0, prove that E is a measurable set.
3. Suppose that E ⊂ Rd is any set and > 0. Prove that there is an open set O
containing E so that m∗ (O) ≤ m∗ (E) + .
4. Recall that E4F denotes the symmetric difference of E and F – the set of
points that lie in exactly one of E, F . Suppose that E ⊂ Rd is well-approximated by
measurable sets in the following sense: for any > 0, there is a measurable set F so
that m∗ (E4F ) < . Prove that E is measurable.
5. Let I ⊂ R denote the set of irrational numbers. Find a way to decompose I
into two sets, C and S, so that C is closed and m∗ (S) < 1/10. (The letter C stands
for closed, and the letter S stands for small. By a decomposition, I mean that C and
S are disjoint sets with union I.)
(As we will learn on Friday, any measurable set can be decomposed into a closed
set and a small set. More formally, if E is measurable and > 0, then E = C ∪ S
with C closed, m∗ (S) < and C and S disjoint.)
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